Kari Astala

Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Area distortion of quasiconformal mappings

In recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane. We show here that, in turn, methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings. It has been known since the work of Ahlfors [A] and Mori [Mo] that K-quasiconformal mappings are locally H51der continuous with exponent 1/K. The function

Calderóns' Inverse Problem for Anisotropic Conductivity in the Plane

Abstract We study the inverse conductivity problem for an anisotropic conductivity σ ∈ L ∞ in bounded and unbounded domains. Also, we give applications of the results in the case when two sets of local boundary data are given.

Beltrami operators in the plane

We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form fz = H(z, fz), h ∈ L(C), where H is a measurable function satisfying |H(z, w1)−H(z, w2)| ≤ k|w1−w2| and k is a constant k < 1. We will also establish the precise invertibility and spectral properties in Lp(C) for the operators I − Tμ, I − μT, and T − μ, where T is the Beurling transform. These operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions. In particular, we prove invertibility in Lp(C) whenever 1 + ‖μ‖∞ < p < 1 + 1/‖μ‖∞. We also prove related results with applications to the regularity of weakly quasiconformal mappings.

Mappings of BMO–bounded distortion

This paper can be viewed as a sequel to the work [9] where the theory of mappings of BMO–bounded distortion is developed, largely in even dimensions, using singular integral operators and recent developments in the theory of higher integrability of Jacobians in Hardy–Orlicz spaces. In this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions. The planar case was studied earlier by G. David [4]. In particular he obtained existence theorems, modulus of continuity estimates and bounds on area distortion for mappings of BMO–distortion (in fact, in slightly more generality). We obtain similar results in all even dimensions. One of our main new results here is the extension of the classical theorem of Painleve concerning removable singularties for bounded analytic functions to the class of mappings of BMO bounded distortion. The setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem, or “the measurable Riemann mapping theorem”, which is not available in higher dimensions. We give a construction to show our results are qualitatively optimal. Another surprising fact is that there are domains which support no bounded quasiregular mappings, but admit

Extremal mappings of finite distortion

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that

Burkholder integrals, Morrey’s problem and quasiconformal mappings

f_o

On the bounded compact approximation property and measures of noncompactness

is a homeomorphism of the circle with

Asymptotic Variance of the Beurling Transform

f_{o}^{-1} \in {\cal W}^{1/2, 2}

Bilipschitz and quasiconformal rotation, stretching and multifractal spectra

. Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on